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Holomorphic Symplectic Manifolds
Marco Andreatta
joint project with J. Wisniewski
Dipartimento di Matematica
Universita di Trento
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Definitions
An holomorphic symplectic manifold Xis a kähler manifold X with a holomorphic non degenerateclosed form σ ∈ H0(X,Ω2
X)An irreducible holomorphic symplectic manifold Xis compact and H0(X,Ω∗
X) = C[σ](eq. X is simply connected and < σ >= H0(X,Ω2
X)).
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Definitions
An holomorphic symplectic manifold Xis a kähler manifold X with a holomorphic non degenerateclosed form σ ∈ H0(X,Ω2
X)An irreducible holomorphic symplectic manifold Xis compact and H0(X,Ω∗
X) = C[σ](eq. X is simply connected and < σ >= H0(X,Ω2
X)).
Calabi-Yau : projective mfds with H0(X,Ω∗X) = C+ Cω,
where ω is a generator for KX .
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Definitions
An holomorphic symplectic manifold Xis a kähler manifold X with a holomorphic non degenerateclosed form σ ∈ H0(X,Ω2
X)An irreducible holomorphic symplectic manifold Xis compact and H0(X,Ω∗
X) = C[σ](eq. X is simply connected and < σ >= H0(X,Ω2
X)).
Calabi-Yau : projective mfds with H0(X,Ω∗X) = C+ Cω,
where ω is a generator for KX .Complex Tori : Cn/Γ, Γ < Cn a discrete subgroup.
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Definitions
An holomorphic symplectic manifold Xis a kähler manifold X with a holomorphic non degenerateclosed form σ ∈ H0(X,Ω2
X)An irreducible holomorphic symplectic manifold Xis compact and H0(X,Ω∗
X) = C[σ](eq. X is simply connected and < σ >= H0(X,Ω2
X)).
Calabi-Yau : projective mfds with H0(X,Ω∗X) = C+ Cω,
where ω is a generator for KX .Complex Tori : Cn/Γ, Γ < Cn a discrete subgroup.
Theorem (Bogomolov). Z kähler manifold with c1(Z) = 0.Up to an etale cover Z ′ ≃ Tori x Calabi-Yau x Irr. Sympl.
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Kummer construction
Following Kummer, Fujiki, Beauville, we takeData:*) A a complex torus of dimension d*) G < GL(r.Z) an irreducible representation of a finitesubgroup; if d is odd we assume G < SL(r.Z).
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Kummer construction
Following Kummer, Fujiki, Beauville, we takeData:*) A a complex torus of dimension d*) G < GL(r.Z) an irreducible representation of a finitesubgroup; if d is odd we assume G < SL(r.Z).Construction:i) Consider the action of G on Ar and take the quotientvariety Y := Ar/G.ii) Take a crepant resolution π : X → Y .
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Kummer construction
Following Kummer, Fujiki, Beauville, we takeData:*) A a complex torus of dimension d*) G < GL(r.Z) an irreducible representation of a finitesubgroup; if d is odd we assume G < SL(r.Z).Construction:i) Consider the action of G on Ar and take the quotientvariety Y := Ar/G.ii) Take a crepant resolution π : X → Y .Remark: Crepant resolution means KX ≃ π∗KY .It is hard to find!!!
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Kummer construction
Following Kummer, Fujiki, Beauville, we takeData:*) A a complex torus of dimension d*) G < GL(r.Z) an irreducible representation of a finitesubgroup; if d is odd we assume G < SL(r.Z).Construction:i) Consider the action of G on Ar and take the quotientvariety Y := Ar/G.ii) Take a crepant resolution π : X → Y .Remark: Crepant resolution means KX ≃ π∗KY .It is hard to find!!!Final Output: A manifold with KX ≃ OX and H1(X,C) = C,i.e. Calabi-Yau or Symplectic.
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Examples
More specifically one can check
*) Finite subgroups of SL(2,Z) acting on A2 = (C/Γ)2.The quotient has rational double points, so there existcrepant resolution, we get Kummer surfaces (K3).
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Examples
More specifically one can check
*) Finite subgroups of SL(2,Z) acting on A2 = (C/Γ)2.The quotient has rational double points, so there existcrepant resolution, we get Kummer surfaces (K3).
*) Finite subgroups of SL(3,Z) acting on A3 = (C/Γ)3.There exist crepant resolutions (Roan and others), we getCalabi-Yau 3-folds.
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Examples
More specifically one can check
*) Finite subgroups of SL(2,Z) acting on A2 = (C/Γ)2.The quotient has rational double points, so there existcrepant resolution, we get Kummer surfaces (K3).
*) Finite subgroups of SL(3,Z) acting on A3 = (C/Γ)3.There exist crepant resolutions (Roan and others), we getCalabi-Yau 3-folds.*) Finite subgroups of Sp(2n,C) acting on An = (C2/Γ)n.due to Fujiki-Beauville.Together with two sporadic examples of O’ Grady they arethe only examples of Irreducible Symplectic mfds.
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Local symplectic contractions
Consider a local symplectic contractions π : X → Y where
X is a symplectic manifold
Y is an affine normal variety,
π is a birational projective morphism with connectedfibers.
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Local symplectic contractions
Consider a local symplectic contractions π : X → Y where
X is a symplectic manifold
Y is an affine normal variety,
π is a birational projective morphism with connectedfibers.
In dimension 2 symplectic contractions are classical andthey are minimal resolutions of Du Val singularities An, Dn,E6, E7, E8.They are quotients of type C2/H with H < SL(2,C) a finitegroup.
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Quotient symplectic singularities
For example take G < Sp(2n,C), i.e. G preserves asymplectic form σ.For any resolution π : X → C2n/G the form π∗(σ) extendsto a holomorphic two form on X (Beauville).If it is non degenerate everywhere then π is a symplecticresolution (a symplectic contraction). This is equivalent tobe crepant.
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Quotient symplectic singularities
For example take G < Sp(2n,C), i.e. G preserves asymplectic form σ.For any resolution π : X → C2n/G the form π∗(σ) extendsto a holomorphic two form on X (Beauville).If it is non degenerate everywhere then π is a symplecticresolution (a symplectic contraction). This is equivalent tobe crepant.Problem: describe G < Sp(2n,C) which admit a symplecticresolution (even for n = 2).
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Examples
Let S be a smooth surface. Then
Hilbn(S) := S[n]S → (S)n/σn := Sn(S)
is a crepant resolution; it is the blow-up of the diagonal
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Examples
Take H < SL(2,C) and let S → C2/H be the minimaldesingularization (symplectic contraction).
Consider the composition
S[n]S → Sn(S) → Sn(C2/H)
it is a crepant map.
It is the symplectic resolution of Sn(C2/H) = C2n/G whereG = (H)n ⋊ σn < Sp(2n).
This is the local Hilbn case of Beauville and Fujiki.
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Examples
Consider the composition S[n+1](C2) → Sn+1(C2) → C2,where the last is τ : (a1, ..., an+1), (b1, ..., bn+1) → (Σai,Σbi).
The restriction X := π−1(0, 0) → τ−1(0, 0) is a crepant map.
It is the symplectic resolution of τ−1(0, 0) = C2n/G whereG = σn+1 < Sp(2n).
This is the local Kumn case of Beauville and Fujiki.
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Symplectic resolution for Cn ⊕ Cn∗
Let G < GL(n,C) a finite subgroup. G can be viewed as asubgroup G < Sp(Cn ⊕ Cn∗) = Sp(C2n) (the symplecticform preserved is the identity in Cn ⊕ Cn∗.
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Symplectic resolution for Cn ⊕ Cn∗
Let G < GL(n,C) a finite subgroup. G can be viewed as asubgroup G < Sp(Cn ⊕ Cn∗) = Sp(C2n) (the symplecticform preserved is the identity in Cn ⊕ Cn∗.Theorem. (Bellamy) Let G < GL(n,C); a symplecticresolution of C2n/G exists iff G is one of the two groupsabove or G is the binary tetrahedral group, BT = Q⋊ Z3,acting on C4 = C2 ⊕ C2∗.
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Symplectic resolution for Cn ⊕ Cn∗
Let G < GL(n,C) a finite subgroup. G can be viewed as asubgroup G < Sp(Cn ⊕ Cn∗) = Sp(C2n) (the symplecticform preserved is the identity in Cn ⊕ Cn∗.Theorem. (Bellamy) Let G < GL(n,C); a symplecticresolution of C2n/G exists iff G is one of the two groupsabove or G is the binary tetrahedral group, BT = Q⋊ Z3,acting on C4 = C2 ⊕ C2∗.Remark. a)Lehn-Sorger described explicitly a localsymplectic resolution for C4/BT .
b) Kummer construction applies for the first two but it does
NOT apply for BT (i.e. there is no global symplectic resolu-
tion) (– ,Wisniewski).Holomorphic Symplectic Manifolds – p.10/35
Special properties
Remarks: Y has rational singularities, KY is trivial and π iscrepant. All exceptional fibres are uniruled.
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Special properties
Remarks: Y has rational singularities, KY is trivial and π iscrepant. All exceptional fibres are uniruled.Theorem (Wierzba - Namikawa) π is semismall, i.e. forevery Y ⊂ X2codim(F ) ≥ codim(π(F )).If equality holds then F is called a maximal cycle.
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Special properties
Remarks: Y has rational singularities, KY is trivial and π iscrepant. All exceptional fibres are uniruled.Theorem (Wierzba - Namikawa) π is semismall, i.e. forevery Y ⊂ X2codim(F ) ≥ codim(π(F )).If equality holds then F is called a maximal cycle.
Theorem (Z. Ran- Wierzba) Let f : P1 → X be a nonconstant map whose image is a π exceptional curve. Thenf deforms in a family (Hilb) of dimension at least dimX + 1.
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McKay correspondence
Let G < Sp(V ) and for g ∈ G : age(g) = 1/2codim(V )g.
Theorem (Batyrev-Kaledin- ...).Assume there exists a symplectic resolution π : X → V/G,then:dimH2i(X,Q) = ♯conj. classes of el. of G of age = i;odd homology is zero.
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McKay correspondence
Let G < Sp(V ) and for g ∈ G : age(g) = 1/2codim(V )g.
Theorem (Batyrev-Kaledin- ...).Assume there exists a symplectic resolution π : X → V/G,then:dimH2i(X,Q) = ♯conj. classes of el. of G of age = i;odd homology is zero.
Moreover there exists a base of H2i(X,Q) given bymaximal cycles which are counter-image of (V )g withcodimV g = i .
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4-dimensional case
From now on we restrict to the case dimX = 4.
By semi-smallness Exc(π) consists (possibly) of
Dj exceptional divisors mapping to surfacesπ(Dj) := Si ⊂ Y
Tk two dimensional special fibersπ(Tk) = pt := 0 ∈ Y .
Remark. (Wierzba)The general fiber F is a tree of rational curves.Components of a general fiber are callled essential curves;if two stay in the same component Dj they are deforamtionequivalent.The normalization of a two dimensional fiber is a rationalsurface. Holomorphic Symplectic Manifolds – p.13/35
Small contractions
Theorem(Wierzba-Wisniewski, Cho-Myiaoka- Sheperd Barron).If π is small (i.e. Dj = ∅) then the Ti are a finite numbers ofdisjoint P2 with normal bundle T ∗P2.(Mukai flop)
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Chow Scheme
Let C ⊂ π−1(s), s ∈ Si \ = 0 be an essential curve andV ⊂ Chow(X/Y ) an irreducible component, i.e. a Chowfamily of rational curve, containing it and such that the mapV → Si → Si is dominant.
Consider the incidence diagram: U
q// Dj ⊂ X
V // Si// Si
where Si is the normalization.
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Chow Scheme
U
q// Dj ⊂ X
V // Si// Si
Proposition (Wierzba, Conde-Wisniewski, — - Wisniewski)V is smooth and it has a holomorphic closed two form nondegenerate (possibly) outside some (−1)-curves.In particular Si has at most a double point singularity at 0and V is a not necessarly minimal desingularization of Si.q is not of maximal rank on the locus over the (−1)-curve.
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Mori Dream Spaces
Let π : X → Y be a projective morphism of normalvarieties with connected fibers and Y = SpecA.By N ef(X/Y ) ⊂ N 1(X/Y ) we understand the closure ofthe cone spanned by the classes of relatively-amplebundlesBy Mov(X/Y ) ⊂ N 1(X/Y ) we understand the conespanned by the classes of linear systems which have nofixed components.
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Mori Dream Spaces
Assume that X is Q-factorial and Pic(X/Y ) is a lattice(finitely generated abelian group with no torsion); letN 1(X/Y ) = Pic(X/Y )⊗Q.We say that X is a Mori Dream Space (MDS) over Y if:
1. N ef(X/Y ) is the affine hull of finitely manysemi-ample line bundles:
2. there is a finite collection of small Q-factorialmodifications (SQM) over Y , fi : X − → Xi such thatXi −→ Y satisfies the above assumptions andMov(X/Y ) is the union of the strict transformsf ∗i (N ef(Xi)).Xi are called the SQM models.
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More on MDS
Note that a version of the theorems of Hu-Keel works in therelative situation too.In particular, the relative Cox ring, Cox(X/Y ), is a welldefined, finitely generated, graded module⊕
L∈Pic(X/Y ) Γ(X,L).Moreover X is a GIT quotient of Spec(
⊕L∈Pic(X/Y ) Γ(X,L))
under the Picard torus Pic(Y/X)⊗ C∗ action.
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Symplectic contractions are MDS
Theorem. Let π : X → Y be a 4-dimensional localsymplectic contraction.Then X is a Mori Dream Space over Y .Moreover any SQM model of X over Y is smooth and anytwo of them are connected by a finite sequence of Mukaiflops.In particular, there are only finitely many non isomorphic(local) symplectic resolution of Y .
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Symplectic contractions are MDS
Theorem. Let π : X → Y be a 4-dimensional localsymplectic contraction.Then X is a Mori Dream Space over Y .Moreover any SQM model of X over Y is smooth and anytwo of them are connected by a finite sequence of Mukaiflops.In particular, there are only finitely many non isomorphic(local) symplectic resolution of Y .Proof
Cone theorem holds (Mori, Kawamata)
Base point free theorem (Kawamata, Shokurov)
Existence of flops (Wierzba-Wisniewski)
Termination (Matsuki). Holomorphic Symplectic Manifolds – p.20/35
Movable cone
N1(X/Y ) denotes the vector space of 1-cycles proper overY . We define Ess(X/Y ) as the convex cone spanned bythe classes of curves which are not contained in π−1(0)
Theorem (— -Wisniewski, Altmann-Wisniewski)Mov(X/Y ) =< Ess(X/Y ) >
∨
.In particular the cone Mov(X/Y ) is symplicial.
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Movable cone
N1(X/Y ) denotes the vector space of 1-cycles proper overY . We define Ess(X/Y ) as the convex cone spanned bythe classes of curves which are not contained in π−1(0)
Theorem (— -Wisniewski, Altmann-Wisniewski)Mov(X/Y ) =< Ess(X/Y ) >
∨
.In particular the cone Mov(X/Y ) is symplicial.
Theorem The subdivision of Mov(X/Y ) into the subconesN ef(Xi/Y ) is done by hyperplanes, corresponding tosmall contractions.(that is the internal walls are of the type C ∩Mov(X/Y )where C is an hyperplane in N 1(X/Y )).
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Semi direct product
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Semi direct product
The resolution X → C4/(Z2)2 ⋊ Z2 contains:
a reducible divisor D = D0 ∪D1,the fiber over 0 which is F4 ∪ P2.
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Semi direct product: Chow
The surface S has a singularity of type A1
V0 is the minimal resolution, the curve in U0 over the −2correspond to curves in F4.V1 is also the minimal resolution and the curve in U1 overthe −2 correspond to the splitting curves in F4 ∪ P2.
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Semi direct product: MDS
Pic(X/Y ) = Z2 = and NE(X/Y ) =< e0, e1 − e0 >, wheree1 − e0 is a line in P2.
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Semi direct product: MDS
Pic(X/Y ) = Z2 = and NE(X/Y ) =< e0, e1 − e0 >, wheree1 − e0 is a line in P2.
There are two symplectic resolution of C4/(Z2)2 ⋊ Z2. They
are symmetric and the flop passes from one to the other.
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Semi direct product: resolution
The resolution X → C4/BT contains:a reducible divisor D = D1 ∪D2,
the fiber over 0 which is F4 ∪ F4 ∪ Blp(P1 × P1) ∪ P2 ∪ P2.
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Semi direct product: resolution
A
B1 B2
C1 C2
A
B1 B2
C1 C2
A
B1 B2
C1 C2
A
B1 B2
C1 C2
A
B1 B2
C1 C2
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Semi direct product: MDS
There are five different symplectic resolutions of C4/(Z3)2⋊
Z2.Holomorphic Symplectic Manifolds – p.28/35
Movable cone for (Zn)2 ⋊ Z2
Theorem Let X → C4/(Zn)2 ⋊ Z2 a symplectic resolution.
The exceptional locus consists of n divisor,D0, D1, ...., Dn−1, and (n+ 2)(n− 1)/2 two dimensionalfibers. Let ei be an essential curve in Di.Then Mov(X/Y ) =< e0, e1, ..., en−1 >
∨
.The division of Mov(X/Y ) into Mori chambers is definedby hyperplanes λ⊥
ij for 1 ≤ i ≤ j ≤ n whereλij = e0 − (ei + ...+ ej).
To each chamber it corresponds a symplectic resolution;
they give all the symplectic resolutions.
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σ3 example
The resolution X → C4/σ3
contains an irr. divisor Da fiber over 0 equal to S3.
The surface S is smooth,V is the blow-up of a point,the curves in U over the −1correspond to lines in S3.Pic(X/Y ) = Z
and this is the uniquesymplectic resolutionof C4/σ3.
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Binary tetrahedral
The resolution X → C4/BT contains:a reducible divisor D = D1 ∪D2,the fiber over 0 which is S3 ∪ F4 ∪ Blp(S3) ∪ P2.π1 contracts D1 as above, π2 is the small contraction of P2.
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Binary tetrahedral: Chow
The surface S has a A1 singularity.
V1 and V2 are both non minimal resolution of this singular-
ities. In each case the fiber contains two rational curves
meeting transversally: a (-1) curve, corresponding to curves
in S3 respectively Blp(S3), and a (−3)-curve.
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Binary tetrahedral: MDS
Pic(X/Y ) = Z2 = and NE(X/Y ) =< e1, e2 − e1 >, wheree2 − e1 is a line in P2.
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Binary tetrahedral: resolutions
There are two symplectic resolution of C4/BT . They aresymmetric and the flop passes from one to the other.
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A conjecture
Conjecture Let π : X → Y be a local symplectic contractionon a smooth projective manifold of dimension 4. Assumethat it is elementary, i.e. that Pic(X/Y ) ≃ Z. Then it is oneof the following:
the small contraction.
the symplectic resolution of C4/σ3
the symplectic resolution of C4/Z2.
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